nLab
coherent logic

Coherent logic is a fragment of first-order logic restricted to $\wedge$, $\vee$, $\top$, $\perp$, and $\exists$. Coherent logic can be interpreted in a coherent category.

A good example of a coherent theory that is not algebraic (in any of the usual senses, although it comes from algebra) is the theory of a local ring; a similar example is the theory of a discrete field.